In this paper, we introduce a notion of categorified cyclic operad for set-based cyclic operads with symmetries. Our categorification is obtained by relaxing defining axioms of cyclic operads to isomorphisms and by formulating coherence conditions for these isomorphisms. The coherence theorem that we prove has the form “all diagrams of canonical isomorphisms commute”. Our coherence results come in two flavours, corresponding to the “entries-only” and “exchangeable-output” definitions of cyclic operads. Our proof of coherence in the entries-only style is of syntactic nature and relies on the coherence of categorified non-symmetric operads established by Došen and Petrić. We obtain the coherence in the exchangeable-output style by “lifting” the equivalence between entries-only and exchangeable-output cyclic operads, set up by the second author. Finally, we show that a generalization of the structure of profunctors of Bénabou provides an example of categorified cyclic operad, and we exploit the coherence of categorified cyclic operads in proving that the Feynman category for cyclic operads, due to Kaufmann and Ward, admits an odd version.

中文翻译：

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分类循环操作数

在本文中，我们为具有对称性的基于集合的循环操作数引入了分类循环操作数的概念。我们的分类是通过将循环操作数的定义公理放松为同构并为这些同构制定相干条件来获得的。我们证明的相干定理具有“典型同构的所有图可交换”的形式。我们的一致性结果有两种风格，对应于循环操作数的“仅条目”和“可交换输出”定义。我们在仅条目风格中的连贯性证明具有句法性质，并且依赖于 Došen 和 Petrić 建立的分类非对称操作的连贯性。我们通过“提升”由第二作者建立的仅条目和可交换输出循环操作数之间的等效性来获得可交换输出风格的一致性。